A characterisation for the category of Hilbert spaces

TL;DR

This paper offers a new categorical characterization of real, complex, and quaternionic Hilbert spaces that does not rely on monoidal structure, expanding the understanding of their categorical properties.

Contribution

It provides an alternative categorical characterization of all three types of Hilbert spaces without assuming monoidal structure, including the quaternionic case.

Findings

New characterization includes quaternionic Hilbert spaces.

Eliminates the need for monoidal structure in categorical characterization.

Extends categorical understanding of Hilbert spaces.

Abstract

The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Sol\`er's theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quaternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quaternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.